It's been a tough week for billionaire investor Bill Ackman, and longtime investors are debating the health of his firm, Pershing Square.

News that Ackman sold down his Mondelez position by 20 million shares on Wednesday night, just a day after losing $1 billion dollars on paper in Valeant, raised a flurry of new questions. His fund is down 9 percent this week and 26 percent year-to-date. Pershing Square's value has also declined 47-percent from its high reached just back in July of last year.

In a letter to investors on Wednesday night, Ackman said the Mondelez sale was for "rebalancing," as the position grew outsized, compared to other holdings, like that of Valeant, which have dropped in value. That sparked wild speculation about whether Ackman might have gotten a margin call, or something else.

David Orrell | CNBC

Bill Ackman

Ackman denied that claim to CNBC on Thursday morning, saying, "We don't use margin leverage — never have and never will."

Other investors speculated that Ackman might be preparing for a wave of redemptions prompted by his portfolio's recent declines, which he said was not the case. Ackman said just 2 percent of Pershing Square's assets, which total more than $11 billion — was redeemed on Feb. 15, the most recent opportunity to do so. Much of the negative Valeant news was already widely known at that time, though the shares had not yet taken the biggest plunge. As of Feb. 15, Ackman said, some 83 percent of his investors who could have cashed out instead rolled their money ahead and remained in the fund.

One long-term Ackman investor told CNBC on Thursday that he has no desire to leave. "We're sticking with him for sure" the investor said, pointing to Ackman's long-term track record, which he called stellar.

He also pointed out successful bets in the Howard Hughes Corp., Air Products and several others as reasons to remain optimistic. The investor further blamed bad press for much of Ackman's ills. "Generally speaking, it's been all negative and there's been a piling on effect as a result."

The next redemption opportunity for Pershing investors won't come around until mid-May, Ackman noted, but he says he's optimistic his investors will stay in the fund, going into detail about why the firm's financial position is solid.

Ackman said Pershing Square has large cash balances in every one of its funds — averaging 14 to 15 percent throughout the portfolio.

He also pointed to three sources of so-called "permanent capital," or money that can't be redeemed — $4 billion dollars from his publicly traded vehicle, Pershing Square Holdings, which is traded in Amsterdam; $1 billion dollars from a recent bond offering; and just more than a billion or so in what Ackman described as "employee capital." He called other money "long-term lock-up capital" where investors remain highly restricted in how they can redeem their money. As is common in many hedge funds – it could take Ackman's investors up to two-years to get all of their cash back.

On the issue of Valeant – where Ackman has suffered steep losses lately — he admitted the challenges ahead but said he could help fix the company — "Valeant is a problem, but we know what to do," he said.